UDC 517.946

MATHEMATICS

A. M. NAKHUSHEV

ON A PROBLEM OF A. V. BITSADZE

(Presented by Academician M. A. Lavrent’ev, 28 X 1969)

Let \(\Omega\) be a bounded simply connected domain of the three-dimensional Euclidean space of points \((x,y,z)\), bounded by a piecewise smooth surface \(\sigma\), situated in the half-space \(z>0\), and by two surfaces \(S_1\): \(x+x_0=\sqrt{y^2+z^2}\) and \(S_2\): \(x_0-x=\sqrt{y^2+z^2}\), \(x_0>0\), lying in the half-space \(z<0\); \(\Omega_1=\Omega\cap(z<0)\), \(\Omega_2=\Omega\cap(z>0)\); \(S_0\) is the part of the plane \(z=0\) separating \(\Omega_1\) from \(\Omega_2\); \(\partial\Omega\) is the boundary of \(\Omega\).

In the domain \(\Omega\) consider the three-dimensional Lavrent’ev–Bitsadze equation

\[ Lu\equiv \operatorname{sign} z\cdot u_{xx}+u_{yy}+u_{zz}=f(x,y,z), \tag{1} \]

which is elliptic for \(z>0\), hyperbolic for \(z<0\), and parabolically degenerates for \(z=0\). The surfaces \(S_1\) and \(S_2\) are characteristics of this equation.

Problem of A. V. Bitsadze. Find a function \(u(x,y,z)\), continuous in the closed domain \(\overline{\Omega}\), with first-order derivatives continuous inside \(\Omega\), satisfying equation (1) in the domain \(\Omega\) for \(z\ne0\) and the boundary condition

\[ u|_{\sigma}=0,\qquad u|_{S_1}=0 \tag{2} \]

or

\[ u|_{\sigma}=0,\qquad u|_{S_2}=0. \tag{2*} \]

We shall adopt the following notation: \(W_2^k(\Omega)\) is the Sobolev space with norm \(\|\cdot\|_k\) and scalar product \(\langle\cdot,\cdot\rangle_k\), \(k=0,1\); \(W(W^*)\) is the set of all functions \(u\) of the class \(C(\overline{\Omega})\cap C^2(\Omega\setminus S_0)\cap W_2^1(\Omega)\cap W_2^1(\partial\Omega)\), for which \(Lu\in W_2^0(\Omega)=L_2(\Omega)\) and condition (2) \(((2^*))\) is satisfied; \(n=(x_n,y_n,z_n)\) is the unit outward normal to \(\partial\Omega\), and \(n^*=(-x_n,y_n,z_n)\) is the conormal corresponding to the Lorentz operator \(\square\equiv -\partial^2/\partial x^2+\partial^2/\partial y^2+\partial^2/\partial z^2\); \(lu\equiv au+bu_x+cu_y+du_z\) is a first-order differential expression with coefficients \(a(x,y,z)\in C^1(\overline{\Omega})\cap C^2(\overline{\Omega}_1)\cap C^2(\overline{\Omega}_2)\), \(b(x,y,z), c(x,y,z), d(x,y,z)\in C(\overline{\Omega})\cap C^1(\overline{\Omega}_1)\cap C^1(\overline{\Omega}_2)\); \(A\) is the set of all four-component vector functions \((a,b,c,d)\) which, in the domain \(\Omega\) for \(z\ne0\), satisfy the system of differential inequalities

\[ La>0,\qquad \operatorname{sign}z\cdot b_y+c_x=0,\qquad \operatorname{sign}z\cdot b_z+d_x=0,\qquad c_z+d_y=0; \tag{3} \]

\[ b_x-c_y+d_z>2a,\qquad b_x+c_y-d_z>2a,\qquad d(x,y,z)=0; \tag{4} \]

\[ \operatorname{sign}z\cdot(c_y+d_z-b_x-2a)>0,\qquad \forall(x,y,z)\in\overline{\Omega}, \tag{5} \]

and the boundary conditions

\[ a=0,\qquad d=0,\qquad bx_n\pm cy_n\le0,\qquad b^2\ge c^2,\qquad \forall(x,y,z)\in S_2. \tag{6} \]

The set \(A\) is nonempty. It contains, for example, vectors whose components have the form

\[ a=\varepsilon\,[y^2+z^2-(x-x_0)^2],\qquad b=b_1(x-x_0)+b_0,\qquad c=c_1y,\qquad d=d(z), \tag{7} \]

where \(\varepsilon, b_1, b_0, c_1\) are arbitrary constants such that

\[ \varepsilon>0,\qquad b_0\leqslant 0,\qquad c_1>|2a|,\qquad b_1>c_1+|2a|,\qquad \forall(x,y,z)\in\overline{\Omega}, \tag{8} \]

and the function \(d(z)=0\) for \(z\leqslant 0\) and satisfies the differential inequality

\[ 2a+b_1-c_1<d'(z)<-2a+b_1+c_1 \tag{9} \]

for \(z\geqslant 0\). For \(c_1=1,\ b_1=\lambda,\ 1<\lambda<2\), and sufficiently small positive values of \(\varepsilon\), the function \(d=z\) is a solution of relation (9).

The inequalities (6) follow directly from (7) and (8), if one takes into account that on the characteristic surface \(S_2\) we have \(\sqrt{2x_n}=1,\ \sqrt{2}(x_0-x)y_n=y,\ \sqrt{2}(x_0-x)z_n=z\). The validity of relations (3), (4), and (5) is obvious under the conditions (8) and (9).

Below we shall assume that the piecewise smooth surface \(\sigma\) has the property that, at least for one vector \((a,b,c,d)\in A\), almost everywhere on \(\sigma\) the inequality

\[ n\cdot(b,c,d)=bx_n+cy_n+dz_n\geqslant 0 \tag{10} \]

holds.

A priori estimate. For any function \(u\in W\) the energy inequality

\[ \|u\|_1\leqslant C\|Lu\|_0, \tag{11} \]

holds, where \(C\) is a constant independent of \(u\).

Indeed, for any function \(u\in W\) and any vector \((a,b,c,d)\in A\), the identity

\[ \begin{aligned} 2\langle lu,Lu\rangle_0 &=2\langle lu,\Delta u\rangle_0+2\langle lu,\square u\rangle_0 \\ &=2\int_{S_2} u^2\frac{\partial a}{\partial n^*}\,dS =\int_{\Omega}u^2La\,d\Omega +\int_{\sigma}\bigl[(bx_n-cy_n-dz_n)u_x^2 \\ &\quad+(-bx_n+cy_n-dz_n)u_y^2 +(-bx_n-cy_n+dz_n)u_z^2 +2(cx_n+by_n)u_xu_y \\ &\quad+2(dx_n+bz_n)u_xu_z +2(dy_n+cz_n)u_yu_z\bigr]\,dS \\ &\quad+\int_{S_1\cup S_2}\bigl[(-bx_n+cy_n+dz_n)u_x^2 +(-bx_n+cy_n-dz_n)u_y^2 \\ &\quad+(-bx_n-cy_n+dz_n)u_z^2 +2(-cx_n+by_n)u_xu_y \\ &\quad+2(-dx_n+bz_n)u_xu_z +2(dy_n+cz_n)u_yu_z\bigr]\,dS \\ &\quad+\int_{\Omega}\bigl[\operatorname{sign} z\cdot(c_y+d_z-b_x-2a)u_x^2 +(b_x-c_y+d_z-2a)u_y^2 \\ &\quad+(b_x+c_y-d_z-2a)u_z^2\bigr]\,d\Omega =I_1+I_2+I_3+I_4+I_5,\qquad I_3=I_3(S_1)+I_3(S_2). \end{aligned} \]

Since \(a=0\) on \(S_2\), and, as is well known, the conormal \(n^*\) lies on the characteristic \(S_2\), it is obvious that \(I_1=0\). The surface \(\sigma\) is a level surface; consequently, on it \(u_x=u_nx_n,\ u_y=u_ny_n,\ u_z=u_nz_n\). Taking this into account, it is easy to see that

\[ I_3=\int_{\sigma}(bx_n+cy_n+dz_n)|n|^2u_n^2\,dS =\int_{\sigma}(b,c,d)\,n u_n^2\,dS, \]

from which, on the basis of (10), we conclude that \(I_3\geqslant 0\).

In a completely analogous way we obtain

\[ I_3(S_1)=\int_{\sigma}(b,c,d)\,n\,(y_n^2+z_n^2-x_n^2)\,u_n^2\,dS=0. \]

The matrix \(M\) of the quadratic form under the sign of the integral \(I_3(S_2)\),

by virtue of (5) has the form

\[ M=\left\| \begin{array}{ccc} cy_n-ax_n & by_n-cx_n & bz_n\\ by_n-cx_n & cy_n-bx_n & cz_n\\ bz_n & cz_n-cy_n & -bx_n \end{array} \right\|. \]

Relying exclusively on the fact that on the characteristic \(S_2\), \(y_n^2+z_n^2=x_n^2\), it is not difficult to show that, when inequality (6) is satisfied, all the principal minors of the matrix \(M\) are nonnegative: \(\det M=0\), \((cy_n-bx_n)^2-(by_n-cx_n)^2=(b^2-c^2)z_n^2\), \(-(c^2y_n^2-b^2x_n^2)-c^2z_n^2=(b^2-c^2)x_n^2\), \(-(c^2y_n^2-b^2x_n^2)-b^2z_n^2=(b^2-c^2)y_n^2\).

Consequently, by Sylvester’s criterion, \(I_3(S_2)\geqslant 0\). Thus,

\[ \int_{\sigma}\left[u^2La+\operatorname{sign} z\cdot(c_y+d_z-b_x-2a)u_x^2+(b_x-c_y+d_z-2a)u_y^2+\right. \]

\[ \left.+(b_x+c_y-d_z-2a)u_z^2\right]\,d\Omega \leqslant \varepsilon\|u\|_1^2+C_1\|Lu\|_0^2, \tag{12} \]

where \(\varepsilon\) is an arbitrarily small positive number, and \(C_1\) is a positive constant independent of \(u\).

From the energy inequality (12) and conditions (3), (4), (5), (11) follows.

In exactly the same way one proves the validity of the a priori estimate (11) for any function \(u\in W^*\).

Inequality (11) generalizes the a priori estimate obtained by A. V. Bitsadze in paper (¹).

By a simple integration by parts one can show that

\[ \langle u,Lv\rangle_0=\langle v,Lu\rangle_0,\qquad \forall u\in W,\quad v\in W^*, \]

and therefore problems (1)—(2) and (1)—\((2^*)\) are (formally) mutually adjoint.

From the a priori estimate (11) there follows the uniqueness of the regular (classical) solution \(u\in W\) or \(v\in W^*\) of A. V. Bitsadze’s problem.

For \(\sigma=S^*\), where \(S^*\) consists of the two conical surfaces \(S_3:\ x_0-x=\sqrt{y^2+z^2}\) and \(S_4:\ x+x_0=\sqrt{y^2+z^2}\), this fact was first established in (¹).

Let now the right-hand side \(f(x,y,z)\) of equation (1) belong to the Hilbert space \(W_2^{-1}(\Omega)\) with negative norm \(\|\cdot\|_{-1}\) and with scalar product \(\langle\cdot,\cdot\rangle_{-1}\).

A weak solution of A. V. Bitsadze’s problem will be any function \(u\in L_2(\Omega)\) satisfying the equality

\[ \langle u,Lv\rangle_0=\langle f,v\rangle_{-1},\qquad \forall v\in W^*. \]

The proof of existence of a weak solution is carried out according to the usual scheme (see, for example, (²), p. 152; (³), p. 107), which we reproduce for convenience of reading.

According to the a priori estimate (11), valid for any function \(v\in W^*\), the expression \(\langle f,v\rangle_{-1}\) depends not on \(v\), but on \(Lv\), and therefore one may set \(\langle f,v\rangle_{-1}=F(Lv)\), where \(F\) is a homogeneous and additive functional on the linear set \(L(W^*)\). Further, on the basis of (11) and the generalized Schwarz inequality, we have

\[ |F(Lv)|=|\langle f,v\rangle_{-1}|\leqslant \|f\|_{-1}\|v\|_1\leqslant C\|f\|_{-1}\|Lv\|_0, \]

i.e., for fixed \(f\in W_2^{-1}(\Omega)\), the functional \(F(\varphi)\), \(\varphi=Lv\), on \(L(W^*)\) is continuous. Extending \(F(\varphi)\) by the well-known Hahn—Banach theorem to the whole space \(L_2(\Omega)\) and using the Riesz theorem, we find the desired function \(u\): \(F(\varphi)=\langle\varphi,u\rangle_0\), \(\forall\varphi\in L_2(\Omega)\), and, in particular, for \(\varphi=Lv\), \(\langle Lv,u\rangle_0=F(Lv)=\langle f,v\rangle_{-1}\).

The energy inequality (11)

\[ \|u\|_1 \leq C\|Lu\|_0, \qquad \forall u \in W \cup W^* \]

also ensures the existence of a semistrong solution of A. V. Bitsadze’s problem (see (³), pp. 98–99).

Among the works devoted to the study of boundary-value problems for an equation of mixed type in multidimensional bounded domains, one should note the work (⁴), where a problem of the Dirichlet-problem type is studied.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk

Received
20 X 1969

REFERENCES

¹ A. V. Bitsadze, DAN, 143, No. 5, 1017 (1962).
² A. V. Bitsadze, Equations of Mixed Type, Publishing House of the Academy of Sciences of the USSR, Moscow, 1959.
³ Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators, Kiev, 1965.
⁴ G. D. Karatoprakliev, DAN, 188, No. 6 (1969).